The emerging field of algebraic statistics advocates the use of polynomial algebra as a tool for statistical analysis. The underlying principle is that many natural families of probability distributions on discrete random variables are algebraic varieties (the zero sets of a collection of polynomials). Knowing the polynomials which define these families of probability distributions can be useful for making statistical inferences and provides a different view point for some problems in probability theory. I will illustrate these ideas with some examples from the theory of graphical models and data security, phylogeny reconstruction, and the theory of conditionally specified models. No knowledge of polynomial algebra will be assumed.